One thing to remember with blogs, when they say "max settings!!!!@" what they really mean is that the game will run above 30 fps at max settings and is thus considered "playable". This is usually true unless they specifically note 60+fps etc. For your purposes below, assume I mean 60+ fps steady.
You can definitely run BF3. Not on literal max settings, but on some very good looking settings, almost surely all "high" settings with post processing on "high" and no MSAA (personally I prefer bf3 with no multisampling anyway as it causes a hazing effect when combined with post processing).
Skyrim is much the same story. Overclocking your CPU and adding the SKSE mod+Tessellation Layer Acceleration plugin for skse, will yield dramatic improvements to that game's performance and you'll probably be able to play on a mixture of high-medium settings without multisampling at most resolution (barring 1920x1200 and beyond).
Crysis 2 similar story as the above 2 games although probably run on a mixture of 2nd and 3rd highest settings for constant 60+
Well for radeon cards the overclocking is Core clock and Mem clock. Shaders function is attached to core clock ratio in all ati cards afaik. SKSE is a skyrim specific mod that you can find on skyrimnexus, and you need it + the tess layer acceleration plugin installed correctly to help compensate for some awful coding in the game regarding cpu performance.
That said, the limits of each card are different and the best thing you can do is look for what overclocks people have achieved stable with a 6870 and start slowly working your way up toward them.
Note: If you mean 1080p as in 1920x1200, your gpu will struggle in BF3 without at least 1.28 and preferably 1.5g VRAM. 1920x1080 though you should be looking pretty good. Could be better but could be much worse.
For overclocking ATI cards though I actually much prefer to completely uninstall Catalyst Control Center, and use strictly ATI Tray Tools. In my experience it was just more convenient and because it let you freely adjust voltage up to the bios limit without any issues, once upon a time I was able to get my 5770 up to 1ghz core clock stable in all games.